Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries includes the Weyl-Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl-Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves invariant the Heisenberg commutation relations are essentially projective representations of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of Hamilton's equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup

Timing of progression from Chlamydia trachomatis infection to pelvic inflammatory disease: a mathematical modelling study

PMCID: PMC3505463The electronic version of this article is the complete one and can be found online at: http://www.biomedcentral.com/1471-2334/12/187. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Helping out: a national survey of volunteering and charitable giving

This report details the main findings of a national survey of volunteering and charitable giving – termed Helping Out – carried out by the National Centre for Social Research (NatCen) in partnership with the Institute for Volunteering Research (IVR) in 2006/07. The study was carried out for the Office of the Third Sector in the Cabinet Office. The main aims of the study were to examine: - how and why people give unpaid help to organisations, and what they think of their experiences; - what stops people from giving help; - the links between giving time and giving money; - how, why and how much people give money to charity; - what stops people from giving money to charity. There was also interest in estimates of the prevalence of volunteering and charitable giving. However, for a number of reasons (detailed in Chapters 2 and 10), prevalence estimates derived from this study should not be used to look at changes in these measures over time. Other study series are better suited to this purpose. In terms of volunteering, the study focused on formal help given through groups and organisations rather than informal help (given as an individual, e.g. to family and friends)

A redshift survey of IRAS galaxies

Results are presented from a redshift survey of all 72 galaxies detected by IRAS in Band 3 at flux levels equal to or greater then 2 Jy. The luminosity function at the high luminosity end is proportional to L sup -2, however, a flattening was observed at the low luminosity end indicating that a single power law is not a good description of the entire luminosity function. Only three galaxies in the sample have emission line spectra indicative of AGN's, suggesting that, at least in nearby galaxies, unobscured nuclear activity is not a strong contributor to the far infrared flux. Comparisons between the selected IRAS galaxies and an optically complete sample taken from the CfA redshift survey show that they are more narrowly distributed than those optically selected, in the sence that the IRAS sample includes few galaxies of low absolute blue luminosity. It was also found that the space distributions of the two samples differ: the density enhancement or IRAS galaxies is only approx. 1/3 that of the optically selected galaxies in the core of the Coma cluster

Mineral Processing by Short Circuits in Protoplanetary Disks

Meteoritic chondrules were formed in the early solar system by brief heating of silicate dust to melting temperatures. Some highly refractory grains (Type B calcium-aluminum-rich inclusions, CAIs) also show signs of transient heating. A similar process may occur in other protoplanetary disks, as evidenced by observations of spectra characteristic of crystalline silicates. One possible environment for this process is the turbulent magnetohydrodynamic flow thought to drive accretion in these disks. Such flows generally form thin current sheets, which are sites of magnetic reconnection, and dissipate the magnetic fields amplified by a disk dynamo. We suggest that it is possible to heat precursor grains for chondrules and other high-temperature minerals in current sheets that have been concentrated by our recently described short-circuit instability. We extend our work on this process by including the effects of radiative cooling, taking into account the temperature dependence of the opacity; and by examining current sheet geometry in three-dimensional, global models of magnetorotational instability. We find that temperatures above 1600 K can be reached for favorable parameters that match the ideal global models. This mechanism could provide an efficient means of tapping the gravitational potential energy of the protoplanetary disk to heat grains strongly enough to form high-temperature minerals. The volume-filling nature of turbulent magnetic reconnection is compatible with constraints from chondrule-matrix complementarity, chondrule-chondrule complementarity, the occurrence of igneous rims, and compound chondrules. The same short-circuit mechanism may perform other high-temperature mineral processing in protoplanetary disks such as the production of crystalline silicates and CAIs.Comment: 6 pages, 3 figures, ApJL published versio

The geometry of reaction norms yields insights on classical fitness functions for Great Lakes salmon.

Life history theory examines how characteristics of organisms, such as age and size at maturity, may vary through natural selection as evolutionary responses that optimize fitness. Here we ask how predictions of age and size at maturity differ for the three classical fitness functions-intrinsic rate of natural increase r, net reproductive rate R0, and reproductive value Vx-for semelparous species. We show that different choices of fitness functions can lead to very different predictions of species behavior. In one's efforts to understand an organism's behavior and to develop effective conservation and management policies, the choice of fitness function matters. The central ingredient of our approach is the maturation reaction norm (MRN), which describes how optimal age and size at maturation vary with growth rate or mortality rate. We develop a practical geometric construction of MRNs that allows us to include different growth functions (linear growth and nonlinear von Bertalanffy growth in length) and develop two-dimensional MRNs useful for quantifying growth-mortality trade-offs. We relate our approach to Beverton-Holt life history invariants and to the Stearns-Koella categorization of MRNs. We conclude with a detailed discussion of life history parameters for Great Lakes Chinook Salmon and demonstrate that age and size at maturity are consistent with predictions using R0 (but not r or Vx) as the underlying fitness function

Temperature Fluctuations driven by Magnetorotational Instability in Protoplanetary Disks

The magnetorotational instability (MRI) drives magnetized turbulence in sufficiently ionized regions of protoplanetary disks, leading to mass accretion. The dissipation of the potential energy associated with this accretion determines the thermal structure of accreting regions. Until recently, the heating from the turbulence has only been treated in an azimuthally averaged sense, neglecting local fluctuations. However, magnetized turbulence dissipates its energy intermittently in current sheet structures. We study this intermittent energy dissipation using high resolution numerical models including a treatment of radiative thermal diffusion in an optically thick regime. Our models predict that these turbulent current sheets drive order unity temperature variations even where the MRI is damped strongly by Ohmic resistivity. This implies that the current sheet structures where energy dissipation occurs must be well resolved to correctly capture the flow structure in numerical models. Higher resolutions are required to resolve energy dissipation than to resolve the magnetic field strength or accretion stresses. The temperature variations are large enough to have major consequences for mineral formation in disks, including melting chondrules, remelting calcium-aluminum rich inclusions, and annealing silicates; and may drive hysteresis: current sheets in MRI active regions could be significantly more conductive than the remainder of the disk.Comment: 16 pages, 13 figures, ApJ In Press, updated to match proof

Modification of Projected Velocity Power Spectra by Density Inhomogeneities in Compressible Supersonic Turbulence

(Modified) The scaling of velocity fluctuation, dv, as a function of spatial scale L in molecular clouds can be measured from size-linewidth relations, principal component analysis, or line centroid variation. Differing values of the power law index of the scaling relation dv = L^(g3D) in 3D are given by these different methods: the first two give g3D=0.5, while line centroid analysis gives g3D=0. This discrepancy has previously not been fully appreciated, as the variation of projected velocity line centroid fluctuations (dv_{lc} = L^(g2D)) is indeed described, in 2D, by g2D=0.5. However, if projection smoothing is accounted for, this implies that g3D=0. We suggest that a resolution of this discrepancy can be achieved by accounting for the effect of density inhomogeneity on the observed g2D obtained from velocity line centroid analysis. Numerical simulations of compressible turbulence are used to show that the effect of density inhomogeneity statistically reverses the effect of projection smoothing in the case of driven turbulence so that velocity line centroid analysis does indeed predict that g2D=g3D=0.5. Using our numerical results we can restore consistency between line centroid analysis, principal component analysis and size-linewidth relations, and we derive g3D=0.5, corresponding to shock-dominated (Burgers) turbulence. We find that this consistency requires that molecular clouds are continually driven on large scales or are only recently formed.Comment: 28 pages total, 20 figures, accepted for publication in Ap